# Convolution of a discrete continuous function for reconstruction

I am confused as to what this book's description is saying exactly:

As with discrete convolution, we can put bounds on the sum if we know the filter’s radius, $$r$$, eliminating all points where the difference between $$x$$ and $$i$$ is at least $$r$$:

The error comes from this sum

They say that points at $$x-r$$ being integer are discarded from the sum, of which they are not if the ceiling takes them in. This must be an error and it has been confusing me for so long now.

## 1 Answer

The statement is:

eliminating all points where the difference between $$x$$ and $$i$$ is at least $$r$$

so eliminate those points for which

$$|x - i | \gt r \tag{1}$$

Isn't that correct? One could argue whether the words say

$$|x - i | \ge r$$

instead, which might be problematic at the border, but using (1) the statement seems OK to me.

The text after the equation says:

Note, that if a point falls exactly at distance $$r$$ from $$x$$ (i.e., if $$x − r$$ turns out to be an integer), it will be left out of the sum.
This is in contrast to the discrete case, where we included the point at $$i − r$$.

So the discrete case includes it the boundary but the continuous case doesn't.

• Yeah that makes sense , however they go onto say if x-r in the algorithm is an integer, then it is left out. This is wrong clearly no. Is this explanation of the reconstruction algorithm just flawed in this simple example i assume. Oct 19 '21 at 15:54
• @simonbalfe Not really. In the continuous case (which is what they're talking about), inclusion or exclusion of the boundary has little impact on the calculation. Inclusion or exclusion in the discrete case does, which is why they include it there.
– Peter K.
Oct 19 '21 at 16:04
• so in sense it doesnt matter , and they just randomly chose to decide that its not included in short Oct 20 '21 at 0:04